Research Areas
My research is focused on the analysis of the nonlinear PDEs with the primary focus on equations arise in fluid dynamics and wave theory. I am mainly interested in the existence, uniqueness and regularity of the PDEs arise in this field. I also address some quantitative properties of solutions to these kind of equations, namely: long time behavior as well as stability. In my research, I apply a wide range of advanced techniques in harmonic analysis and PDE's, including Littlewood--Paley theory. In addition, I make use of the methods in dynamical systems, and my recent works on both decay rates and stability of PDE's rely heavily on the spectral theory.
In this context, I have worked on the Navier-Stokes systems and wave theory. Since their derivation, many physical phenomena are modeled by wave equations and Navier-Stokes systems. Navier-stokes systems are usually derived by coupling the Navier-Stokes equations with other equations. For example the Magnetohydrodynamics (MHD) system consists of a coupling of the Navier-Stokes equation with the Maxwell's equation, and models the magnetic properties of electrically conducting fluids, including plasma, liquid metals, electrolytes, etc.
Projects in preparation
Global regularity for the 2D Magneto-Micropolar Equation in the super critical region
With With J. Kelliher and D. Ambrose: Surface Quasi_geostrophic Patches and Layers.
With D. Ambrose: Global Well-posedness and long time behavior of the coordinate free model of the flame front.
With D. Wright: The dynamic of diatomic FPU system toward monotonic FPU system.
Publications
With A. Stefanov: Sharp Asymptotics for global solutions of the parabolic Hardy-Sobolev equation. (Submitted)
With H. Mofidi: On the sharp time decay rates for the fractional Boussinesq system. (Submitted)
With J. Kelliher and D. Ambrose: Generalized Surface Quasi_geostrophic Patches and Layers. (To appear in Nonlinearity)
With H. Mofidi: Analysis of Flux Ratios Bifurcation in Ionic Flows via Classical Poisson-Nernst-Planck Models, Studies in Applied Mathematics (2025)
With David Ambrose and J. Kelliher: Periodized solutions to the 2D Euler equations, periodic arrays of vortex patches, and periodic vortex layers. Siam J. on Mathematical Analysis (2024).
With F. Farivar: A Chemotaxis model of Prey and Predator model of three species: Existence of global solutions., JMAA (2024)
F. Hadadifard, Optimal decay rates and attractors of the 2D fully dissipative Magneto Hydrodynamics system., ZAMP, 72, Article number: 132 (2021).
With D. Ambrose, D. Wright, Well-posedness and asymptotics of a coordinate-free model of flame fronts. Siam J. on Applied Dynamical system, (2021).
With A. Stefanov, Steady states of the weakly forced SQG equation: existence and sharp relaxation rates. J. Math. Fluid Mechanics, (2021).
With D. Wright, Mass-in-Mass Lattices with Small Internal Resonators. Studies in Applied Mathematics, (2021).
With A. Stefanov, Sharp relaxation rates for plane wave of general reaction-diffusion models. J. Math. Phys. Vol.61, Issue 4, (2020).
With S. Malhi, Z. Xiao, C_0-Semigroup Numerical Approximation to Damped Wave equations., Journal of Numerical Analysis, Industrial and Applied Mathematics, ( 2020).
(Ph.D Thesis) With A. Stefanov, Sharp time asymptotics for the quasi-geostrophic equation, the Boussinesq system and near plane waves of reaction-diffusion models.
With A. Stefanov, On the sharp time decay rates for the 2D generalized Quasi-Geostrophic equation and the Boussinesq system. Journal of Nonlinear Science, 29(5), (2019), pp 2231-2296.
With A. Stefanov, On the global regularity of the 2D critical Boussinesq system with $\alpha > 2/3$, Comm. Math. Sci., 15(2017), p. 1325-1351.
With H.Mofidi, Weight guaranteeing polefree barycentric rational interpolation, Indian J. of Science and Technology, 6(11) (2013), p. 5450-5458.